My questions mostly concern the history of physics. Who found the formula for kinetic energy
$E_k =\frac{1}{2}mv^{2}$
and how was this formula actually discovered? I've recently watched Leonard Susskind's lecture where he proves that if you define kinetic and potential energy in this way, then you can show that the total energy is conserved. But that makes me wonder how anyone came to define kinetic energy in that way.

My guess is that someone thought along the following lines:

Energy is conserved, in the sense that when you lift something up
you've done work, but when you let it go back down you're basically
back where you started. So it seems that my work and the work of
gravity just traded off.
But how do I make the concept mathematically rigorous? I suppose I need functions $U$ and $V$, so that the total energy is their sum $E=U+V$, and the time derivative is always zero, $\frac{dE}{dt}=0$.

But where do I go from here? How do I leap to either

a) $U=\frac{1}{2}mv^{2}$

b) $F=-\frac{dV}{dt}$?

It seems to me that if you could get to either (a) or (b), then the rest is just algebra, but I do not see how to get to either of these without being told by a physics professor.

We can thank Émilie du Châtelet for the modern, unified understanding Kinetic energy. You are right that the full theory of Energy conservation was difficult. It started with Liebniz and didn't end until Noether.
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user121330Aug 27 '14 at 21:02

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Addem (and others), you may also be interested in this SE proposal: History of Science
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DanuAug 28 '14 at 13:54

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I just want to point out that equation a that you use is not correct. U is generally used as a symbol for potential energy, but the rest of the equation provides the way to calculate kinetic energy.
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SeanOct 6 '14 at 13:33

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@BenCrowell, " It's basically not true that this...." and "There is no "energy formula." . These are apodictical propositions, Ben, and too heavy statements to be just dropped in a couple of comments, without one link to substantiate them. You can write an answer and document your point of view any time
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abcNov 5 '14 at 6:33

7 Answers
7

As you probably know, Newton thought that energy is linearly proportional to velocity: the Latin terms vis [force] and potentia [potence, power] were used at that time to refer to what today is called energy.
The second law's original formulation reads: "Mutationem motus proportionalem esse vi motrici impressae" = "any change of motion (velocity) is proportional to the motive force impressed".

This law, which nowadays is wrongly interpreted as: $F = ma$ (there is no reference to mass here) simply states states: $$[\Delta/\delta v]( v_1-v_0) \propto Vis_m$$ and in modern terms is sometimes (illegitimately) also interpreted as impulse, sort of : $$\Delta v \propto J [/m] \rightarrow \Delta p = J$$. But mass is not at all mentioned in the second law (as the original text shows) but only in the second definition, where we can see a definition of momentum as 'the measure of [quantity of] motion'

and, moreover 'motive force' (vis motrix) is used, like all other scholars of the time, referring to the yet unknown kinetic 'force' that made bodies move, which Galileo had called 'impeto' and Leibniz 'motive power'. The interpretation of this formula as the definition of force in modern usage is an ex post facto historical manipulation, done against the author's own will: he knew about this interpretation proposed by Hermann and refused to adopt it in the final edition

The historical facts

It was Gottfried Leibniz, as early as 1686 (one year before the publication of the Principia) who first affirmed that kinetic energy is proportional to squared velocity or that velocity is proportional to the square root of energy: $$ v \propto \sqrt{V_{viva}}$$. He called it, a few years later, vis viva = 'a-live/living' force in contrast with vis mortua = 'dead' force: Cartesian momentum ([mass/weight =] size * speed: $m *|v|$). This was accompanied by a first formulation of the principle of conservation of kinetic energy, as he noticed that in many mechanical systems of several masses $m_i$ each with velocity $v_i$,

$\sum_{i} m_i v_i^2$

was conserved so long as the masses did not interact. The principle represents an accurate statement of the approximate conservation of kinetic energy in situations where there is no friction or in elastic collisions. Many physicists at that time held that the conservation of momentum, which holds even in systems with friction, as defined by the momentum:
$\,\!\sum_{i} m_i v_i$
was the conserved kinetic energy.

The concept of PE played no role, it did not exist yet, nor did the concept of mechanical energy to which you refer (E = U + V), but Leibniz, in this first paper, uses the term potentia motrix/ viva [motive power] to refer both to the energy a body acquires falling from an altitude and to the force necessary to lift it to the same altitude (mass/weight * space: $F*s$) which are considered equal. Some scholars see, wrongly, here a first definition of PE, but that is simply one of the axioms of Galileo.

The principle to which you refer: $E_{mech} [KE + PE] = k$ in astrodynamics is called vis viva equation in his honour. Leibniz stated the conservation of KE per se besides the conservation of all (kinds of) energy in the whole universe. We need to underline this amazing stroke of genius.

His theory was strongly adversed by Newton[ians] and DesCartes-ians because it seemed to contrast, to be incompatible with the conservation of momentum. In Newton there was no distinction (as shown above) between speed, motion, momentum and energy but quantitas motus (momentum) was the prevailing concept and it was proven to be conserved in all situations, therefore Leibniz' vis viva was considered a threat to the whole system. Only later it was acknowledged that both energy and momentum, being different entities, could be conserved (by Bošković and later (1748) by d'Alembert).

We can thank Émilie du Châtelet for the modern..understanding of kinetic energy – user121330

There is no energy formula ..in the discovery of
conservation of energy are Joule and... – Ben Crowell

That's overlooking historical facts (Joule was not concerned with KE): soon after Leibniz' death, the quadratic relation was confirmed by experiments independently by the Italian Poleni in 1719 and the Dutch Gravesande in 1722, who dropped balls from varying heights onto soft clay and found that balls with twice speed produced and indentation four times deeper. The latter informed M.me du Châtelet of his results and she publicized them. Two centuries later, after Joule had shown that mechanical work can be transformed in heat, Helmholz suggested that the lost energy, in inelastic collisions, might have been transformed in heat.

Thomas Young is thought to have been the first to substitute the terms 'vis viva/ potentia motrix' with 'energy' in 1807 (from the Greek word: ἐνέργεια energeia, which had been coined by Aristotle on the stem of ergon = work, therefore: energeia [= the-state-of-being-at-work]). Later (1824-1829) Coriolis introduced the current formula and the terms 'work' and 'semi-vis viva'; this concept and the consequent theory of conservation of energy was eventually formalized by Lord Kelvin, Rankine et al. in the field of thermodynamics.

The formula of kinetic energy

The question is much more complex than it appears, as there are at least four formulas involved here, and each issue is complex in its turn:

how, when and by whom was the formula for the second law of motion $F=m*a$ introduced

how was the formula of kinetic energy $V_{viva} = [m]* v^2$ found by Leibniz

how, when and by whom was the current newtonian formula of kinetic energy $E_k = [m]*\frac{v^2}{2} $ introduced

how, when and by whom was the formula for work $W = F*d$ introduced

I did not want to make this post too long, but I'll take the suggestion from the bounty and address the issues in separate answers. Just a brief note here to make this post self-contained: the formula of KE was not derived from work, as it may seem: it's the other way round. $W = F * d$ and $F = m * a$ were by-products of the KE formula. Once the quadratic relation had been verified and universally accepted: $E \propto v^2$, any coefficient (0.2, 0.5, 2..) could be added as an irrelevant and arbitrary choice that depended only on the choice of units.

The only avalaible (and precisably measurable) source of KE at the time was gravity and the Galilean equations were too strong a temptation, as they included, too, a [0.5] quadratic relation: it seemed a stroke of genius to make the energy of the unitary mass at unitary (uniform) acceleration coincide with space. In this way energy was simply the integration of [m] $g$ on space.

Conclusions

Tying energy to gravity, that is, to acceleration and in particular to constant acceleration was not a wise idea, it was a gross mistake that tied, confined newtonian mechanics in a strait-jacket because it was in this way unable to deal with the more natural situations when KE is related to velocity and when there is just a transfer of energy: the concept of impulse was just an ad hoc awkward attempt to deal with that.

Tying work-energy to space and not to the mere transfer of energy was an insane decision that had irrational, catastrophic practical consequences. But consequences were even more devastating on the conceptual, theoretical level because explaining and identifying KE with the acceleration gave the illusion that the issue of motion-KE had been understood, and prevented further speculation.

Leibnizinvented the concept of (kinetic) energy, prefigured and discovered its real formula $E = v^2$ resisting the Siren of gravity, suggested the right way of integration and established the universal principle of 'conservation of energy' as prevailing on/independent from 'conservation of momentum' (transcending Huygens' principle of 'conservation of KE').

He engaged in passionate controversies until his death but was opposed and overwhelmed by obtuse/ignorant Newtonian contemporaries. He was vulnerable as he could not account for the loss of energy in inelastic collisions. He lost, and newtonian integration on space produced: $\frac{1}{2}$ mv2 which is not the formula, but just one of the possible formulas of KE: the newtonian formula. Had he won, instead of the joule, now we would use the 'leibniz' (= 1/2 J) and we would have a different, probably deeper, insight into the laws of motion and of the world.

Momentum and energy are two ways of quantifying motion. I don't think it's correct to say that Newton was talking about energy at all; he simply defined a momentum as a convenient measure of the movement power of motion.
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Larry HarsonAug 28 '14 at 16:16

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@LarryHarson: " I don't think it's correct to say that Newton was talking about energy at all" Right, and who said that? I wrote 'he made no disctinction'
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abcAug 28 '14 at 16:19

I haven't accepted an answer until now because I just couldn't understand this answer--it sounds very well-informed and probably does answer the question, but I can't tell, because it's just not stated in a way that's quite accessible to me. But I noticed there's a bounty that's about to expire and figured it probably would suck to have put this much effort into an answer without reward, so I'll just accept it anyway.
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AddemDec 25 '14 at 19:07

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@Addem, your OP asked both : "who found the formula for (kinetic) energy" and "who found the formula $\frac{mv^2}{2}$" , which (I hope you realize by now) are 2 different questions and require 2 separate answer, (which become 4 if you want to know "how" ). The new posts address the separate questions, and I hope they are accessible
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abcDec 26 '14 at 7:41

Note. When I wrote the main post I wasn't able to find the book by Coriolis. There is no English translation and the original French text was nowhere to be found in google-books: my explanation of the KE formula was a mere logical deduction. When you read the original work, you realize that it is really tailor-made on gravity. Force (m*a) is simply represented by P[oids] ( i.e. weight) = mg, and energy becomes simply $\int P\delta s$. Also, in the original title the word 'effet' does not make much sense in modern French, same as in the English translation: "Calculation on the 'effect' of the machines".(see here)

In 1826 the French engineer Prony in his paper "Rapport sur la machine du Gros-Caillou" introduced «l’unité dynamique française», the French unit of measurement of the work of the machines in substitution of the British HorsePower defined by Watt and Boulton at the end of the 18th century, as 33,000 ft*lbf per minute. A few years before that, in 1819, Navier had already proposed to the term 'quantity of action' for the metric unit: $Kg*m$, which had been favourably received by Coulomb. In the same year Gaspard-Gustave de Coriolis in a paper submitted to the Académie des sciences proposed the term dynamode for the unit: 1000 Kg*m per minute.

This was not a great innovation, the problem was only to find a name for the equivalent of the BHP in the metric system . In 1829 Coriolis published his book "Du calcul de l’effet des machines" [1], in which he criticizes (for various reasons) the terms which geometers in use at the time:"effet dynamique, "puissance mécanique" and "quantité d'action" and proposes to call the unit: "travail [dynamique]": 'dynamic work' or simply work, in which the modifier 'dynamic' is intended to give the word a more technical connotation and distinguish it from everyday usage. [1, p.14-17].

But he says that he doesn't want to change the term in use for the kinetic energy: force vive [vis viva] ($V_v = \frac{p}{g}* v^2$), but, for the sake of simplicity in the enunciation of the principles he will consider the ['semi vis viva' = ] demi-force vive ($ DFV = \frac{pv^2}{2g}$), because it corresponds to the height from which a body must fall to get that energy.:

His proposals were universally accepted, but for the 'name': in 1889 during the Universal exposition in Paris, the International Congress of Electricity named his unit the 'joule', substituting the old electrical unit of volt*coulomb.

There is no originality in his proposal, he did not 'find' the formula of KE, but there are two 'revolutionary' changes in his work:

the introduction of the concept of work, as a substitute of the concept of power, and its units, with the same units of KE, to describe the 'effect' of a machine

the modification of Leibniz' formula of the relation between velocity and energy, to adapt it to the force of gravity

The Galilean laws included a quadratic relation with reference to space [height/ distance]: $d = 0.5 * a*t^2$, It was simple to make the energy of the unitary mass at unitary (uniform) acceleration coincide with space: $ [m, a = 1] \rightarrow E = d = [1*]\frac{t^2}{2}$.

I intended to divide the bounty among the three answers, but after I clicked this post it was impossible to go on. I am particularly sorry about the third answer, which has not been rewarded in any way.
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user59485Dec 27 '14 at 7:18

I suspect, though I'm not sure, that a nineteenth century French mathematician and scientist, Gaspard-Gustave Coriolis, is your man. He was the first to define the notion of "work done" and even kinetic energy. His wiki reads:

In 1829 Coriolis published a textbook: Calcul de l'Effet des Machines ("Calculation of the Effect of Machines"), which presented mechanics in a way that could readily be applied by industry. In this period, the correct expression for kinetic energy, ½mv2, and its relation to mechanical work, became established.

I guess that many mathematicians in that era independently found a ½mv2 formula based on Coriolis's work, though it seems likely that Coriolis was the first.

You've already got some answers, but nobody mentioned Noether's Theorem yet. Noether's theorem maps a conserved quantity to each continuous symmetry. The relevant continuous symmetry needed to prove the conservation of energy is the one that leaves the laws of nature invariant, meaning the laws of physics don't change with time. Each continuous symmetry implies a certain function and the time derivative of that function must be zero. If you wanna read more about it, check out the wikipedia entry or any book about classical mechanics!
Noether's Theorem on wikipedia.

Note: from the invariance of space (the laws of physics are the same everywhere in space) momentum conservation follows!

Noether's theorem was way way after Leibniz.
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Love LearningAug 28 '14 at 9:43

I'm aware of that, but the OP asked for a mathematically rigorous proof, and Noether's theorem is the closest thing to that.
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user17574Aug 28 '14 at 10:33

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OK, but adding some history would make your answer richer.
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Love LearningAug 28 '14 at 12:35

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@user17574 There are much much simpler proofs, all of these conserved quantities, momentum, energy, angular momentum could be proved rigorously just from Newton's laws. Also considering the Lagrangian involves the kinetic energy, I don't think this helps.
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JLAAug 30 '14 at 7:17

I deliberately ignored this aspect of the question in my main post because that is a very complex conceptual issue, which can't be honestly dismissed with a few lines. On the other hand, giving an adequate answer would have made that post too long and dense and I informed the OP. Moreover, theoretical questions are not particularly appreciated by the general public at this site, anyway I'll take the suggestion of the bounty and try to reach a fair compromise, giving a rather short answer that covers all the main points.

So he just noticed a pattern in a bunch of examples? So he did
something like: Work out the masses and velocities of this and that
system, and just by chance he realized that in each example, if you
square the velocities and dot product it with the masses that you'd
get a constant? – Addem

No description could be more far from truth, when you are talking about Leibniz. He never realized something by chance, he made no experiments: he was no physicists, he simply interpreted 'Galilean laws' in the right, logical way.

He was a philosopher and looked for (and found) absolute truths because of abstract logical, epistemological, metaphysical and even religious reasons and deductions. In that same way the ancient 'sages' from a few fundamental first principles (the object of metaphysics) such as symmetry and proportionality were able to establish fundamental truths: "infinitum in actu non datur", "Natura non facit saltus" and the consequent "horror vacui", abstract laws of geometry or even physical truths, like Eraclitus precisely describing the ultimate structure of matter and chemical bonds with the "atoms in iron, that link themselves to other atoms with a hook", or practical findings such as the measurement of the circumference of the Earth.

In 1686 he published his Brevis demonstratio ("A Brief Demonstration of a Notable Error of Descartes...)" to disprove his theory (upheld by his followers) that the total quantity of motion (size*speed : $m*|v|$) in the universe is conserved

Proposition 11: If two bodies collide with each other, and if the
ratio of both their magnitudes and their speeds is given either in
numbers or in lines, then the sum of their magnitudes multiplied by
the squares of their respective velocities is equal before and after
the collision

but had given no explanation of what was actually conserved. Most likely, Leibniz was not aware of his law (wrongly associated to vis viva), since it was published in Paris three years before is arrival on a diplomatic mission 1672-'76), it is true that he met him there and that from '73 he was in correspondence with him, but, even if he was aware of his law, he made no use of it, reaching his goal through a completely different path.

And the amazing thing is that he used the same path as Coriolis would follow two centuries later (the Galilean laws), getting to a different conclusion.

Descartes' derivation of his principle was transcendental, but naive. He believed (rightly) that the 'vacuum' is impossible (horror vacui) but he concluded (wrongly) that all space is filled with matter, and therefore motion of one body implies the pushing of the body ahead, and so on.

Leibniz believed in the same religious principles, but concluded (rightly) that it is force, potence ( = potentia motrix, Vis viva) that is conserved and not momentum, and in a couple of pages liquidates his opponent's theory:

the Vv a brick (1-A) acquires falling from a certain altitude is equal to Vv required to raise same body to same altitude. Vv 1 brick (1-A) acquires by falling from a height of 4 meters is equal to the amount of Vv a four-brick body (4-B) acquires by falling from a height of one meter. Therefore A will get Vv sufficient to raise 4-B by 1 meter. By Galileo's Law a falling body traveling a distance/height (H) of (1) meter in [1] second will travel (4) meters in [2] seconds and (16) meters in [4] seconds. 1-A, falling from H = 4 will have v = 2 and p = 2, 4-B falling from H = 1 will have v = 1 and p = 4. quantitas motus are not equal but potentia, Vv are equal.

Therefore, he concludes, the force, potence, Vis viva, energy of a body cannot be calculated by finding its 'quantity of motion' (m*v) but is to be estimated from the 'quantity of the effect' [quantitas effectus] it can produce, that is from the height to which it can elevate a body of given magnitude ($V_v = m *v^2$). It is possible to see here an anticipation of the concepts that will named work and potential energy

His transcendental belief made him conclude that the total amount of VisViva in the world is conserved both locally and globally with the result that there is always as much Vv in a cause as in its effect. It is really amazing to read how he justified the loss of energy in collisions of soft bodies (inelastic):

Since every body is composed of infinitely many smaller bodies, it
must be attributed to the fact that 'Vis viva 'has been transferred
to the smaller parts of the whole body in a way that has no
influence on the motion of the whole. It is lost as KE but it is
nonetheless conserved in the sense that it is still present in the
motion of the smaller bodies from which the larger bodies are
constituted: what is absorbed by the minuscule parts (atoms,
molecules) is not absolutely lost for the universe.

Like Eraclitus centuries before, he has exactly anticipated and described what two centuries later Kelvin will formalize. His explanation of the transformation of kinetic energy into heat is no worse than Feynman's explanation of the KE lost by a ball bouncing on the floor with the jiggle of the atoms

Note: about a century later the enlightened positivist d'Alembert banned metaphysics from physics and apparently solved the controversy declaring that both momentum and vis viva can be and are indeed conserved. This is not true because total momentum is not always conserved and, moreover, because before Leibniz the concept of quantity of motion quite often corresponds to the concept of kinetic energy

The responses given so far are fairly accurate however, the question you should be asking goes to the experimental proof for the kinetic energy formula. Mathematically, the formulas for work and kinetic energy seem to function perfectly as taught. Unfortunately, there are at least 2 or 3 situations where it does not. No physics teacher ever looks at these and so, physics students never get the full story.

I'll give you one scenario that no one will argue against. Imagine you are performing a space walk and you throw a wrench. Because the wrench is far less massive than you, it has far more work done to it than you. If instead of a wrench you threw a small satellite that had the same amount of mass as you did and threw it with the same effort as the wrench, the tossed item would have the same amount of work done to it as was done to you. The change in kinetic energy for both you and the thrown items would be different; the total ke for you and the wrench would be far more than the ke for you and the satellite even though you used the same amount of biological energy in both cases.